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Full Idea
Identity can be circularly defined, as 'the relation everything has to itself and to nothing else', …or as 'the smallest equivalence relation'.
Gist of Idea
Identity definitions (such as self-identity, or the smallest equivalence relation) are usually circular
Source
Harold Noonan (Identity [2009], §2)
Book Ref
'Stanford Online Encyclopaedia of Philosophy', ed/tr. Stanford University [plato.stanford.edu], p.2
A Reaction
The first one is circular because 'nothing else' implies identity. The second is circular because it has to quantify over all equivalence relations. (So says Noonan).
Related Ideas
Idea 10104 'Equivalence' is a reflexive, symmetric and transitive relation; 'same first letter' partitions English words [George/Velleman]
Idea 16017 Identity is usually defined as the equivalence relation satisfying Leibniz's Law [Noonan]
| 22322 | You can't define identity by same predicates, because two objects with same predicates is assertable [Wittgenstein] |
| 17594 | We can paraphrase 'x=y' as a sequence of the form 'if Fx then Fy' [Quine] |
| 10797 | Substitutivity won't fix identity, because expressions may be substitutable, but not refer at all [Marcus (Barcan)] |
| 9848 | Content is replaceable if identical, so replaceability can't define identity [Dummett, by Dummett] |
| 9842 | Frege introduced criteria for identity, but thought defining identity was circular [Dummett] |
| 11831 | The formal properties of identity are reflexivity and Leibniz's Law [Wiggins] |
| 16497 | Leibniz's Law (not transitivity, symmetry, reflexivity) marks what is peculiar to identity [Wiggins] |
| 16498 | Identity cannot be defined, because definitions are identities [Wiggins] |
| 16502 | Identity is primitive [Wiggins] |
| 16015 | Problems about identity can't even be formulated without the concept of identity [Noonan] |
| 16017 | Identity is usually defined as the equivalence relation satisfying Leibniz's Law [Noonan] |
| 16016 | Identity definitions (such as self-identity, or the smallest equivalence relation) are usually circular [Noonan] |
| 16020 | Identity can only be characterised in a second-order language [Noonan] |
| 6053 | Identity is as basic as any concept could ever be [McGinn] |